On the Distributional Stieltjes Transformation

This paper is concerned with some general theorems on the distributional Stieltjes transformation. Some Abelian theorems are proved. Throughout the paper, r will denote a positive continuous function on an interval (X,=), X > 0, such that the limit lim r(pt) r(t) exists for' every p > 0. Such functions are called regularly varying fu.Lc-tions (r.v.f.) at infinity and it is well known ([ 7] that they are of the form r(t) taL(t) for some a R (called the order or index of r) and some slowly varying function (s.v.f.) L. This means that the function L The quasiasymptotic behaviour (q.a.b.) at infinity of tempered distributions with support n [O,m) (denoted by S+) ws defined by Zavi-jalov (see, for instance, [2]). In this paper we use a somewhat more general concept of q,a,b, related to a r.v.f, as defined and analysed in Definition 1. Let T S+ and r be some r.v.f. The distribution T has q.a.b, at infinity related to r if there exists the limit in the sense of S" lim T(kt r(k) g(t Provided that g 0.